On the Method of Calculating the Differential Coefficients of a Function from its Differences; and on their application to the Interpolation of functions of one, two or three variables

Extract

The development of the nth differential coefficient of a function in terms of the finite differences is involved in the expansion of the multinomial . A general expression for the coefficient of each power, to the seventh inclusive, is given in De Morgan's Diff. and Int. Calculus, p. 264,—but the extremely intricate character of such general expression renders it desirable to obtain a more convenient practical method of computing these coefficients to any extent that may be required.